Simulated Annealing with two ns

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Simulated Annealing(with two ns)

• by Albert Young-Sun Kim
• Friday, October 21st 2005
• Unsupervised Learning Workgroup (ULWG)

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Analogy What is Annealing?

• A chemical process used to make the strongest
  possible glass.
• (Like the glass surrounding a hockey rink)

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Process of Annealing

• Start by heating glass at high temperature,
  allowing molecules to move freely
• Lower temperature of the glass slowly so that at
  each temperature the atoms can move just enough
  to begin adopting the most stable orientation.
• Do this until temperature no longer alters glass.

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Main Ideas of Annealing

• The temperature determines how much mobility the
  atoms have.
• How slowly you cool glass is critical. This
  rate is called the cooling schedule (aka
  Annealing Schedule).
• If the glass is cooled slowly enough, the atoms
  are able to "relax" into the most stable
  orientation.

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What is Simulated Annealing?

• Simulated Annealing (SA) uses the same ideas as
  regular annealing
• It is a Metropolis Monte Carlo (MMC) Optimization
  Method(used especially when the global extrema
  are hidden amongst many local extrema).

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Digression Metropolis Algorithm

• The scheme of taking downhill steps while
  sometimes taking an uphill step is known as a
  Metropolis Algorithm
• Contrast this with a Greedy Algorithm, which
  would only takes downhill steps.

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Process of SA

• Structure A SA program consists of two
  nested do-loops.
• Outer loop Sets temperature T which determines
  the age of random steps that result in an
  increase of the function that will be accepted
  (See next slide).
• Inner loop Does MMC at that T (See Flow Chart)

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Linchpin of SA

• During an MMC iteration, if at current guess for
  minima X
• Function value goes down, ACCEPT X.
• Function value goes up, ACCEPT X with probability
  
• where
• T is the temperature of the system
• k is the Boltzmann Constant (Metropolis)
• is the difference in
  function value from the previous iteration

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Main Ideas of SA

• Temperature is determines mobility
• Cooling schedule can be linear or proportional.
• For any finite problem, P(SA terminates at Global
  Optima) 1 as annealing schedule is extended
  (process will reach relaxed state).

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Moral of the Story

• Because we also allow for increases (not just
  decreases) in objective function f (with a
  certain probability), we can escape local
  minima.
• i.e. We will never be stuck in a rut!!!

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Pros and Cons

• Pros
• SA's major advantage over other methods is an
  ability to avoid becoming trapped at local
  minima.
• Cons
• How do we determine the cooling schedule?i.e.
  how do we decide what is a sufficient amount of
  iterations at each temperature?

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Main References of SA

• N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth.
  A.H. Teller and E. Teller. Equation of State
  Calculations by Fast Computing Machines J. Chem.
  Phys. 21 (1953) 1087-1092.
• S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi.
  Optimization by Simulated Annealing Science 220
  (1983) 671-680.
• Numerical Recipes in C The Art of Scientific
  Computing 2nd Ed. pp. 444-454